Killer Sudoku Cage Combinations

In killer sudoku each dashed cage shows a total, and no digit may repeat inside a cage. So a cage of two squares with total 4 can only be 1 and 3: 2 and 2 would repeat a digit. Most totals allow several sets of digits, a few allow only one, and knowing which is which is most of the early game.

How to read the tables

• Possible digit sets lists every set of different digits from 1 to 9 that adds up to the total. "136" means 1, 3 and 6 in some order. The table tells you which digits; the rest of the puzzle tells you the order.
• Always has is the digits found in every set. If a cage's total has an "always" digit, that digit is in the cage, wherever the cage's other digits end up.
• Never has is the digits found in no set. You can cross those off every square in the cage straight away.
• Shaded rows have only one possible set. Look for these first.

Three examples

A 3-square cage of 8. The table gives two sets: 1, 2, 5 and 1, 3, 4. Both contain a 1, so the cage holds a 1 for certain. If the cage sits inside one box, no other square in that box can be 1. And 6, 7, 8 and 9 are out of all three squares.

A 3-square cage of 22. The sets are 5, 8, 9 and 6, 7, 9. The 9 is certain. If the whole cage lies inside one row, no other square in that row can be 9, and the same goes for a column or a box.

A 2-square cage of 10. Four sets: 1 and 9, 2 and 8, 3 and 7, 4 and 6. Nothing is always present and only 5 is never present (5 and 5 would repeat). This is the least helpful 2-square total, so leave it until other clues narrow it.

The patterns worth remembering

You do not need to learn the whole table. Four rows per cage size do most of the work: the two smallest and the two largest totals each have only one set.

• 2 squares: 3 = 1+2, 4 = 1+3, 16 = 7+9, 17 = 8+9.
• 3 squares: 6 = 1+2+3, 7 = 1+2+4, 23 = 6+8+9, 24 = 7+8+9.
• 4 squares: 10 = 1+2+3+4, 11 = 1+2+3+5, 29 = 5+7+8+9, 30 = 6+7+8+9.

Totals near the middle of the range have the most sets and tell you least. A 3-square cage of 15 has eight possible sets.

A shortcut for big cages

A cage of 9 squares holds 1 to 9, which add up to 45. So a cage of 6 squares is the same as "the three digits missing from 1 to 9 add up to 45 minus the total". A 6-square cage of 38 leaves out three digits that add to 7, and only 1, 2, 4 add to 7. So the cage is 3, 5, 6, 7, 8, 9. That is why the tables for 6, 7 and 8 squares below mirror the tables for 3, 2 and 1 squares. You can do it in your head: big cage, subtract from 45, read the small table.

The same 45 fact powers the main killer technique, the rule of 45: every row, column and box adds up to 45, so a box covered by cages except for one square gives that square away.

The full tables

A cage of all 9 squares always totals 45 and holds every digit, so it gives no extra information beyond the normal sudoku rules.

Using it while you play

1. Scan the grid for shaded-row totals and single-square cages first. Pencil those digits in with Notes.
2. For every other cage, cross off its "never has" digits. In a puzzle with several small or large totals, this alone removes a lot.
3. Mark the "always has" digit of each cage. Where that cage sits inside a single row, column or box, the digit is fenced in, and the rest of that unit loses it.
4. Then apply ordinary sudoku logic: the techniques page still applies in full.

Ready to try it? Our killer sudoku page generates puzzles that each have exactly one solution, checked by our solver.